14 REPORT — 1859. 



the other in Scotland, and corresponding stations well chosen in America, would 

 enable us to transmit messages across the Atlantic. 



On the Affections of Polarized Light reflected and transmitted by thin 

 Plates. By the Rev. H. Lloyd, D.D., F.R.S. 



When plane- polarized light is incident upon a thin plate, the reflected and trans- 

 mitted pencils are, in general, elliptically polarized. This fact was pointed out by 

 the author many years ago, as a result of theory ; and it appears to furnish the 

 explanation of the phenomena recently observed by Sir David Brewster, and to which 

 he has called the attention of the Members of this Section. In the present commu- 

 nication the author proceeds to deVelope this theoretical result, and to deduce the 

 laws according to which the elliptical polarization varies, as well with the thickness 

 of the plate, as with the incidence. 



When light incident upon a thin plate is polarized either in the plane of incidence, 

 or in the perpendicular plane, it will continue polarized in the same plane, after the 

 successive reflexions and refractions which it undergoes at the two surfaces of the 

 plate ; and we have only to seek the magnitude of the resultant vibration. The 

 problem is different, however, when the light is polarized in any other plane. In this 

 case the incident vibration must be resolved into two, in the two principal planes, 

 and for each of these components we must know the phases, as well as the magnitudes, 

 of the resultant vibrations, before we can estimate their joint effect. As these phases 

 are in general different, the resulting light is elliptically polarized. 



When the media are the same on the two sides of the plate, the difference of phase 

 of the two component vibrations (upon which the character of the resulting light 

 mainly depends) is given by the formula 



(« 2 — w 2 ) sin as 



tan A: 



1 — (v 2 +w 2 ) cos «+v 2 w> 2 ' 



in which aj is the phase due to the retardation of the wave, which has passed once 

 to and fro within the plate ; and v and w the coefficients of the reflected vibrations, 

 for light polarized in the plane of incidence, and in the perpendicular plane, respect- 

 ively. It follows from this, that A varies with et, and therefore with the thickness 

 of the plate ; and that, in the phenomena of the rings, it will yo through all its values 

 within the limits of each ring. 



A vanishes, when a.=.mir, i. e. both at the bright, and at the dark lings ; and 

 accordingly the light at the former is plane-polarized. 



On the other hand, A is a maximum, relatively to the thickness of the plate, when 



v-+w- 

 cos«=- 



'\+v-w- ' 

 and the maximum value is given by the formula 



tan A = 



V(l -«')(! -m> 1 ) 



Substituting for v and w their well-known values in the former of these formula?, 

 we find 



4cot 2 ^=( /;i + / x-i) 2 -|-(p-p->) 2 ; 



m 



in which /* is the refractive index, and p the ratio of the cosines of incidence and 

 refraction. It follows from this, that cot — increases, from — fyi-f/n-^at a perpen- 

 dicular incidence, to infinity when the incidence is most oblique ; and that, in a 

 plate of varying thickness, the points of maximum difference of phase commence 

 near the middle of an interval, and approach indefinitely to the dark rings as the 

 incidence approaches to 90°. 



A similar discussion of the second formula shows that the maximum difference of 

 phase increases continuously with the incidence, being nothing at a perpendicular inci- 



